ABSTRACT
Many of the early concepts of grains and grain shapes were developed from two dimensional models, and were subsequently applied to three dimensional granular arrays, with suitable adaptations for the increased geometrical constraints. For a two dimensional array of polygons, there is a simple relationship between the number of polygons, the number of edges where two polygons meet, and the number of comers where three or more polygons meet. This is given as- https://www.w3.org/1998/Math/MathML"> P − E + C = 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003059417/055db691-660f-45de-b0a2-83501efbe930/content/eqn2_1.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> where P, E, and C are the numbers of polygons (or other flat shapes), edges (lines) and comers (points) respectively. This equation is true irrespective of the number of polygons meeting at a comer, or whether the array includes straight or curved lines. The addition of a point on any edge not only increases the number of points, but also increases the number of edges by unity. The addition of a new curved line between two existing points is the array not only increases the number of edges, but also increases the number of enclosed areas (not necessarily polygons) by unity. This is true even if the curve begins and ends at the same point in the array. These observations ensure compliance with equation 2.1. By way of example, consider the array of shapes shown in Fig. 2.1. The number of points is 57, the number of edges is 44, and the number of polygons or other enclosed shapes is 14, so that, from equ. 2.1- https://www.w3.org/1998/Math/MathML"> 14 − 57 + 44 = 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003059417/055db691-660f-45de-b0a2-83501efbe930/content/eqn0003.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
